3.3.1 Galerkin's Method for the Scalar Differential Operator

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3.3.1 Galerkin's Method for the Scalar Differential Operator

Equation (3.9) is written as

$\displaystyle \int_{\mathcal{V}}N_i \left[\vec{\nabla}\cdot(\utilde{a}\cdot\ve... ...}\right] \mathrm{d}V - \int_{\mathcal{V}}N_if \mathrm{d}V = 0, i\in[1;n],$

(3.14)

where Galerkin's method ( ) and the scalar differential operator (3.2) are used. After applying the first scalar Green's theorem

$\displaystyle \int_{\mathcal{V}}W\left[\vec{\nabla}\cdot(\utilde{a}\cdot\vec{u}... ...m{d}A - \int_{\mathcal{V}}\vec{\nabla}W\cdot\utilde{a}\cdot\vec{u} \mathrm{d}V$

(3.15)

(3.14) is modified to read

$\displaystyle \int_{\partial\mathcal{V}}N_i \vec{n}\cdot\utilde{a}\cdot\vec{\n... ...\tilde{u} \mathrm{d}V - \int_{\mathcal{V}}N_if \mathrm{d}V = 0, i\in[1;n].$

(3.16)

Since vanishes on $\mathcal{A}_D$ for $i\in[1;n]$ the boundary integral from (3.16) reads

$\displaystyle \int_{\partial\mathcal{V}}N_i \vec{n}\cdot\utilde{a}\cdot\vec{\n... ...\nabla}\tilde{u} \mathrm{d}A \simeq \int_{\mathcal{A}_N}N_i u_n \mathrm{d}A.$

(3.17)

In (3.17) the conormal derivative $\vec{n}\cdot\utilde{a}\cdot\vec{\nabla}\tilde{u}$ corresponds with the Neumann boundary condition on $\mathcal{A}_N$ (3.4) [35,37]

$\displaystyle \vec{n}\cdot\utilde{a}\cdot\vec{\nabla}\tilde{u} \simeq u_n = \vec{n}\cdot\utilde{a}\cdot\vec{\nabla}u \mathrm{on} \mathcal{A}_N.$

(3.18)

Equation (3.16) leads with (3.5) and (3.17) to the linear equation system (3.12), where and $\{d\}$ are given by

$\begin{displaymath}\begin{split}K_{ij} & = - \int_{\mathcal{V}}\vec{\nabla}N_i\c... ..._N}N_i u_n \mathrm{d}A, i\in[1;n], j\in[1;n], \end{split}\end{displaymath}$

(3.19)

with $\mathcal{L}[v]$ from (3.2).

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A. Nentchev: Numerical Analysis and Simulation in Microelectronics by Vector Finite Elements